A car is tested along specifically designed track. First, the car is driven along a straight section of track of length (r). If the car starts from rest under a maximum (constant) acceleration, it takes an amount of time (t) to cover the distance (r). The car is then brought to rest, and is continually accelerated along a section of track that makes a circular arc of radius (r). Assuming that the car is speeding up at the maximum possible rate that allows it to remain on the track, how long would it take to traverse the arc?
I don't know how to write equation so I am sub-scripting my variables by attaching _* at the end.
V_avg = distance / time
V_avg = (V_f + V_o) / 2
acceleration = (V_f - V_o) / time
a_centripetal = v^2 / r
The Attempt at a Solution
V_o = 0
a = V_f / time
V_avg = V_f / 2
V_f = 2 * V_avg
V_avg = distance / time = r/t
V_f = 2 * (r/t)
a = 2 * r / t^2
a_centripetal = a (because the car is speeding up at the maximum rate, is this logic correct??)
a_centripetal = (2 * r) / t^2
but a_centripetal = v^2 / r
so v^2 / r = (2 * r) / t^2
solving for v:
v = r/t * sqrt(2)
This is where I am stuck, I found the tangential velocity, but what do I do next?
Say the arc length is created by an angle theta, but theta isn't given, but the arc length would be theta * r, but where would I go next if I go with this route?
Or I could somehow relate the tangential velocity with the angular velocity... but how...??? And I would still need theta right?
I asked my professor and he said that the problem is stated correctly and I don't need to know the coefficient of friction or the arc length.. so I am totally lost >_>
EDIT: I remember posting this problem like a day or two ago but I can't seem to find that thread